A098564 -id:A098564 - cp's OEIS frontend

A098565 Numbers that appear as binomial coefficients exactly 6 times.

120, 210, 1540, 7140, 11628, 24310, 61218182743304701891431482520

Offset: 1

Paul D. Hanna, Oct 27 2004

Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, arXiv:2107.09107 [math.NT], 2021.
Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video, 2020.
David Singmaster, How Often Does An Integer Occur As A Binomial Coefficient?, American Mathematical Monthly, 78(4), 1971, pp. 385-386; also on Fermat's Library.
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

See A098564 for more information.
Cf. A185024, A182237. Subsequence of A003015.
Cf. A059233.

import Data.List (elemIndices)
a098565 n = a098565_list !! (n-1)
a098565_list = map (+ 2 ) $ elemIndices 3 a059233_list
-- Reinhard Zumkeller, Dec 24 2012

A059233(a(n)) = 3. - Reinhard Zumkeller, Dec 24 2012

a(7) from T. D. Noe, Jul 13 2005

A137905 Numbers that appear as binomial coefficients exactly twice.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

David Wasserman, Feb 21 2008

Complement of A006987; a(n) = A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009

			7 is a member because 7 = binomial(7, 1) = binomial(7, 6) and no other binomial coefficient equals 7. [clarified by _Jonathan Sondow_, Jan 12 2018]

Cf. A006987, A058084, A062527, A098564, A098566.

isok(n) = (sum(i=0, n, sum(j=0, i, binomial(i,j)==n)) == 2) \\ Michel Marcus, Jun 16 2013

a(n) = A185024(n+1). - Elijah Beregovsky, May 14 2019

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1210

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

All Narayana numbers A001263(n,k) with n != 2*k-1, are terms since A001263(n,k) = A001263(n,n+1-k). In particular, all positive triangular numbers except 3 are terms. Are there any other terms, i.e., is there a number A001263(2*k-1,k), k >= 2, that can be written as a Narayana number in another way? Any such number would also be a term of A376001.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.

from bisect import insort
from itertools import islice
def A376000_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 2: yield N0
                break
            central = n==2*k-1
            c += 2-central
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376000_list(nmax):
    return list(islice(A376000_generator(),nmax))

A138496 Where record values occur in A003016.

Original entry on oeis.org

0, 1, 10, 120, 3003

Offset: 1

Reinhard Zumkeller, Mar 20 2008

It appears that the record values are 0, 3, 4, 6, 8, 10, 12, ...
From M. F. Hasler, Feb 16 2023: (Start)
The numbers that appear 3 times in Pascal's triangle are the central binomial coefficients (A000984), except for the number 2 that is the only number to appear only once. For n = 1 there would be an infinite number of occurrences, but sequence A003016 counts only the occurrences of n in rows <= n so that n = 1 also gives 3.
All C(n,k) with 1 < k < n/2 (in particular triangular numbers A000217) appear at least 4 times; see A098564 for those appearing exactly 4 times.
Numbers that appear 5 or more times are quite rare, they are listed in A003015 with subsequence A098565 of those appearing exactly 6 times.
They are mostly C(n,k) with 2 < k < n/2 which are also triangular numbers, but some are also of the form C(n+1,k) = C(n,k+1) with 3 < k < n/2, and a subsequence of these has n and k given in terms of Fibonacci numbers. (End)

m=-1; [n | n<-[0..9999], m < m = max(A003016(n), m)] \\ M. F. Hasler, Feb 16 2023

m=-1; [n for n in range(9999)if m < (m := max(A003016(n), m))] # M. F. Hasler, Feb 16 2023

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A098565 Numbers that appear as binomial coefficients exactly 6 times.

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A137905 Numbers that appear as binomial coefficients exactly twice.

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A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

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A138496 Where record values occur in A003016.

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